Arthur C. Clarke / Gentry Lee (click on names to see more mathematical fiction
by the same author)

...

Contributed by
Sara Lioi

This is the sequel to the novel Rendezvous With Rama by Arthur C. Clarke.

Short Summary:
The huge cylindrical Rama spaceship has returned 70 years after it
arrived near Earth for the first time. Another crew is put together and
sent to explore the ship. The crew consists of military and scientific
personnel along with two journalists who are there to report news to
Earth. Almost immediately things start going awry and three crew members
wind up dead. The majority of the crew isn't sure if the Ramans are
friendly or malignant. One crew member, medical officer Nicole des
Jardins, gets separated from the rest of the crew and knocked
unconscious. After being stranded at the bottom of a hole for several
days, one of the biots climbed down into the hole and Nicole hitched a
ride out on its back. She explores the area and makes friends with the
avians, aliens that look sort of like pterodactyls. Eventually another
crew member, engineer and genius Richard Wakefield, finds Nicole and
together they further explore the area known as "New York" (known as this
because of the tall buildings and the fact that it's an island). Finally
they enlist the help of the avians to carry them across the Cylindrical
Sea, and they meet up with General Michael O'Toole, who was taking one
last look around the spaceship before leaving as the rest of the crew had
done. In the end the three of them stay on the Rama ship instead of
returning to Earth.

Mathematical Aspects:

p.79 Richard and Nicole are at a New Year's Eve party at a reconstructed
Roman building. Richard realizes that the building looks very much like the
layout of the Rama ship and comments that the Ramans must know the same
mathematics as the Romans did.

p.174 The Ramans built "cities" inside the spaceship that have buildings
that are all geometrical shapes. Richard recognizes spheres, pentahedrons,
dodecahedrons, etc.

p.254 Dr. David Brown, physicist and astronomer, is noted as being a
purely theoretical scientist because he hated the detail and formality of
empirical science and didn't like engineers or the machines used to prove
theories.

p.312-313 Wakefield describes how he was able to find Nicole because of a
homing device. Using triangulation he was able to find her according to x-y
coordinates but he didn't think about z coordinates (she was underground).

p.352-353 Richard uses geometric relationships in 2 of the 3 sectors of
"New York" to find a third opening to an underground set of tunnels in the
third sector. He used the locations of the other two openings in the other
sectors to figure it out.

p.360 Richard finds an alien computer in one of the underground tunnels
and discovers through math that 1,024 different commands could be used, and
he tries to figure them out.

p.411-412 General O'Toole has to come up with a 50 number code to use on
the nuclear weapons, which minimizes the chance of someone else being able
to crack the code. There is a discussion of the prime numbers 41, 43, 47,
53, 61, 71, 83, 97 and how the difference between the first two is half of
the difference of the second 2, etc. This is the beginning of a sequence of
primes that ends at the number 41x41, or 1681. [See note below. -ak]

p.451 Richard tells O'Toole that no one can know what Rama is all about,
and says that to try and figure it out now would be like solving a system
of simultaneous linear equations when there are many more variables than
constraints. He also says that the entire problem of whether or not to tell
the Raman spaceship that nuclear missiles are headed toward it can be
represented by a 3x2 matrix, and he describes the logic behind the matrix.

Duane Mattingly, wrote to ask me to expand on Sara Lioi's comments about the number 41, which he says "has been my `favorite number' ever since I read Rama II somewhere back on the verge of pubescence". He is correct that one small piece of information has been left out of this description, without which one cannot get the full numerological (and nuministic!) significance of this short sequence. So, please allow me to attempt to explain:

Start with the number 41, a prime number. Add two to get 43, another prime. Add four to that to get the prime 47. Add six and the sum 53 is still prime. Continuing in this way (increasing the difference by two each time), you get 1523 (yup, prime) as your 39th number and by the time you get the prime number 1601 as the fortieth number and you check that it is prime you may have yourself convinced that this is a way to produce an infinite sequence of prime numbers. But, you run into trouble (as Sara explains) and get 1601+80=1681 which is not prime as the next number in the sequence. This in itself is interesting, since it demonstrates the need for proof and rigor in math. But, one cannot help also being impressed by the number 41 after noticing that the sequence which starts with the number 41 stops being prime at the 41st entry with the number 1681=412! [BTW A quick way to find the nth number in this sequence is to use the formula 41+n2-n. This also makes it obvious that the 41st number would be forty-one squared.]

Contributed by
Anonymous

Here's another one:
In chapter 18, Nicole computes the conditional probability of a drug reaction (initially 4%) given that the original diagnosis (appendicitis 92%) is ruled out.

Contributed by
Tina Chang

Personally I loved "Rendevous with Rama" and felt it had much more mathematics than this one. The concept of the centrifugal force is wonderful described there when they first descend the stairs of the ship. It really helped me understand what all those equations in Physics I were about which gave me a stronger understanding of vector calculus. "Songs of a Distant Earth" also by Clarke has a nice intuitive description of vector calculus in the sense of the forces on a space elevator (a cable lowered from a space ship in a geosynchronous orbit, lifting cargo). Maybe these aren't so obviously mathematics as have a few geometric shapes mentioned but this is real mathematics that can help engineers and other vector calculus students really see things. Keep in mind: Clarke's math is correct!

Contributed by
Richard Acton

I thought I would let you know that the discussion of the 41+n^2-1 quadratic primes comes up again the sequel; garden of Rama in this passage in chapter 4:

"Michael told me that 41 was a "very special number." He then explained to me that it was the largest prime that started a long quadratic sequence of other primes. When I asked him what a quadratic sequence was, he laughed and said he didn't know. He did, however, write out the forty-element sequence he was talking about: 41, 43, 47, 53, 61, 71, 83, 97, 113... concluding with the number 1,601. He assured me that every one of the forty numbers in the sequence was a prime. "Therefore," he said with a twinkle, "forty-one must be a magic number."

While I was laughing, our resident genius Richard looked at the numbers and then, after no more than a minute of playing with his computer, explained to Michael and me why the sequence was called "quadratic." "The second differences are constant," he said, showing us what he meant with an example. "Therefore the entire sequence can be generated by a simple quadratic expression. TakeI(A0 = N2 - N + 41," he continued, "where N is any integer from 0 through 40. That function will generate your entire sequence.

"Better still," he said with a laugh, "consider f(N) = N2 - 81N + 1681, where N is an integer running from 1 to 80. This quadratic formula starts at the tail end of your string of numbers, f(1) - 1601, and proceeds through the sequence in decreasing order first. It reverses itself at 40) = f(41) = 41, and then generates your entire array of numbers again in increasing order."